Transcendental arguments for mathematics?

Kant’s unknown world

Kant tells us that due to our conceptual vocabulary; we, in a fundamental way, cannot understand objects as they are in themselves. That is, we only have phenomenal and experiential intuition of the manifold by means of an a priori process of applying the categories to turn bare experience into concepts and framed such into a language to which we can understand; that language (I use the word as a metaphor here) being cognition itself.

The categories are fundamental primitives by which we cannot prove, but cannot go without in our everyday understanding of the world.

Transcendental arguments

A transcendental argument has the form such that we take a given proposition, or better put, we take a proposition as a given; as our starting premise for analysis. Following analysis; we find, if a transcendental argument is to work; that the knowing of the conditions of the proposition’s possibility is such that we concede to some stronger enthymeme that we take for granted, as the bais of understanding the given proposition of analysis.

Let me give an example. You are reading this article. I should hope that this is necessarily true, at least as a contingent proposition indexical to right now. Now, if we analyse this proposition (namely ‘you are reading this article’); we must presuppose, as the conditions of knowing that this proposition is possible, in terms of its truth-aptness, and its modality, and its very cogizance.

You are reading this sentence right now. What does right now mean? Right now is an indexical term, and we cannot have indexicals without an underlying substratum of points of reference insofar to make the reference index ‘right now’ understandable; therefore, there is an underlying substratum of points of temporal reference. QED

Now; here’s the clincher. I didn’t argue to say that space as a real thing, or that space was absolute; I’ll leave that to the Leibnizians and Newtonians to argue about! I just wanted to point out that there are things we must assume to grant that a certain (reasonable) given is true. Some certain reasonable givens are those most basic facts about existence and our everyday lives.

“I am in love” presupposes “a ‘love’ interpersonal relation as a kind of relation between subject and object is possble”; or, “I am running” presupposes “I can run”, or “I have the relevant facility to ‘run’ (namely legs…or otherwise)”

In my honest opinion; I don’t think very much of Transcendental arguments; in fact, modern TAs are possibly the least interesting development to come out of Kant scholarship. But that’s neither here nor there…

Incompleteness

Certain platitudes in early 20thC logic, and to a lesser extent, Set Theory have always come to mind when I think about Kant’s Noumena; or perhaps, as Michael would put it, whenever I think of logic, I think of how it relates to Kant.

I will have more thoughts about Cantor’s project of the continuum hypothesis at a later date; but let me say this. My allegiance for this article is with Michael’s systematicity principle; and in the provoking of discussion of this principle itself; I want to imagine that this certain mathematical platitude I shall sketch may fit into the epistemological issue in Kant.

I’ve found a putative definition of what people call the multiplicative axiom.

A1: For every set A, of non-empty sets x, there exists a choice set

D1: A choice set is a set consisting exactly one member from each x in A

For a certain kind of conventional set theory, this is fairly important; however, due to ‘forcing’ (Cohen), and ‘incompleteness’ (Gödel) [I apologise for my mathematical ‘handwaving’]. We find that this conception of a choice set is neither provable nor refutable from standard axioms of ZF set theory.

Conceptual primitives

My thought, well, this is just speculation, is this. If we may have these odd cases in mathematics where we cannot prove nor refute certain things that we take as given; what say we of our status of them? Both in terms of pragmatic considerations; and metaphysical concerns. I shall phrase this in another way:

Quid facti: should we use these principles?

Quid juris: are we legitimate to use these principles?

Now; here’s the thought: if there are things in the most the foundations of mathematics and logic that are subjects of neither proof or disproof; but neessary for our mathematical discourse; can we give a transcendental argument for it (or in true Kantian terms; can we give a Transcendental Deduction following our Transcendental analytic of principles?)

In other words; can we transcendentally deduce  Axiom of Choice the multiplciative axiom?
Furthermore; is it desirable?

Magisters Destre and Michael, for instance, have heard, of the view that in the interest of parsimony, we may go without this axiom. Furthermore; does this further the systematicity agenda to recode the incompleteness results in Kantian terms? I admit I haven’t said much to argue for the latter view; rather than suggest an example in which it can be played out…

Perhaps one way to express the question is as follows: are there things we cannot prove or disprove; but yet nonetheless assume to guarantee the possibility of knowledge? This is the ultimate Kantian thought.

Sinistre

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One thought on “Transcendental arguments for mathematics?

  1. Magister; you are seriously mistaken

    Kant’s thought about the conditions of knowability do NOT involve mathematical concepts; if anything, perhaps logic.

    Argument 1:

    Let me state the counterfactual; can we imagine mathematical conduct without AC?

    If no, then it is a transcendental assumption
    If yes, then this is NOT a candidate for analysis an analogous way to the pure categories of the understanding!

    Transcendental presuppositions are a claim about the NECESSARY CONDITIONS OF KNOWLEDGE. It is a reasonable claim perhaps to ask whether we cannot but assume Modus Ponens; ut not AC.

    Argument 2:

    A further argument against your thesis is this: can we prove modus ponens through anything other than Transcendental Arguments?

    If no, then it is a candidate for Kantian transcendental analysis analogous to the categories
    If Yes, then it is not…

    Argument 3:

    Set theory is not the foundation of cognition; just (arguably) a possible foundation of mathematics. Do we need the ZFC axioms, or some other set theoretic model like the Von Neumann/Bernays/Godel system presupposed in order to know “this is a cat?”

    I think not, Magister.

    perhaps we make a weak, and non-Kantian claim to the effect that we may presuppose some mathematical platitudes insofar as some given discourse beneath set theory can be made possible; for example, assuming some axiom to justify arithmetic or analysis. Perhaps that is a legitimate thought; but it is hardly Kantian, and hardly an issue for systematic philosophy.

    Remember the quote by DK Lewis; we are supposed to be on the side of science, but not take sides within science.

    Neat argument, Sinistre; but I’m afraid it fails.

    Michael

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